library(dplyr)
library(INLA)
library(inlabru)
library(sf)
library(terra)
# load some libraries to generate nice map plots
library(scico)
library(ggplot2)
library(patchwork)
library(mapview)
library(tidyterra)Practical 6
Aim of this practical:
In this practical we are going to look at space-time model and model comparison and validation techniques.
Space time models
Libraries to load:
The data
In this practical, we will revisit the data on the Pacific Cod (Gadus macrocephalus) from a trawl survey in Queen Charlotte Sound. The pcod dataset is available from the sdmTMB package and contains the presence/absence records of the Pacific Cod during each surveys along with the biomass density of Pacific cod in the area swept (kg/Km\(^2\)). The qcs_grid data contain the depth values stored as \(2\times 2\) km grid for Queen Charlotte Sound.
The dataset contains presence/absence data from 2003 to 2017. To make computations faster we only consider the first 3 years.
library(sdmTMB)
pcod_df = sdmTMB::pcod %>% filter(year<=2005)
qcs_grid = sdmTMB::qcs_gridThen, we create a sf object and assign the right coordinate reference to it:
pcod_sf = st_as_sf(pcod_df, coords = c("lon","lat"), crs = 4326)
pcod_sf = st_transform(pcod_sf,
crs = "+proj=utm +zone=9 +datum=WGS84 +no_defs +type=crs +units=km" )We convert the covariates into a raster and assign the same coordinate reference:
depth_r <- rast(qcs_grid, type = "xyz")
crs(depth_r) <- crs(pcod_sf)Spatio-temporal modeling
Model fitting
Now lets compare two different space-time models using LGOCV and some information criteria metrics. The general model structure is given by:
\[ \begin{aligned} y(s,t)|\eta(s,t)&\sim\text{Binom}(1, p(s,t))\\ \eta(s,t) &= \text{logit}(p(s,t)) \\ \end{aligned} \] We also want to compare the models using WAIC, DIC and marginal likelihood:
Model 1
- Model 1 - time iid effect We consider a separable space-time model with a linear predictors given by:
\[ \eta(s,t) = \beta_0 + f_1(\text{depth}(s)) + f_2(t) + \omega(s) \]
\(f_1(\text{depth}(s))\) is a smooth covariate effect of depth (modeled using a RW2 model)
\(f_2(t)\) is an IID effect of time
\(\omega(s)\) is Matérn random field.
The first step is to define the mesh and the spde model
Construct the mesh and the SPDE model
mesh = fm_mesh_2d(loc = pcod_sf,
cutoff = 1,
max.edge = c(10,20),
offset = c(5,50),
crs = st_crs(pcod_df))
spde_model = inla.spde2.pcmatern(mesh,
prior.sigma = c(1, 0.5),
prior.range = c(100, 0.5))create time index and the grouped variable
To use the RW2 model the covariate has to be groupes:
depth_r$depth_group = inla.group(values(depth_r$depth_scaled))we also define a time index from 1 to in the data frame
pcod_sf = pcod_sf %>%
mutate(time_idx = match(year, c(2003, 2004, 2005)),
id = 1:nrow(.)) # Observation id for CVNote that there are some survey locations in certain years that fall outside the depth raster region. inlabru will input these missing covariate values using the nearest available value. This can be computationally expensive, but you can avoid it by supplying a raster layer that encompasses all of your data points (e.g., by pre-imputing these missing values with your preferred method of choice).
One way of doing this in the inlabru framework is to use the bru_fill_missing() function:
# Select the raster of interest
depth_orig = depth_r$depth_group
re <- extend(depth_orig, ext(depth_orig)*1.05)
# Convert to an sf spatial object
re_df <- re %>% stars::st_as_stars() %>% st_as_sf(na.rm=F)
# fill in missing values using the original raster
re_df$depth_group = bru_fill_missing(depth_orig,re_df,re_df$depth_group)
# rasterize
depth_filled <- stars::st_rasterize(re_df) %>% rast()
plot(depth_filled)We have now fit the model and want to check the results.
Model 2
- Model 2 - spatiotemporal field We consider a separable space time model with a linear predictor given by:
\[ \eta(s,t) = \beta_0 + f_1(\text{depth}(s)) + \omega(s,t) \]
- \(f_1(\text{depth}(s))\) is a smooth covariate effect of depth (RW2)
- \(\omega(s,t)\) is a space-time Matérn spatial field with AR1 time component \[ \omega(s,t) = \phi\ \omega(s,t-1) + \epsilon(s),\qquad \epsilon(s)\sim\text{GF}(\sigma_{\epsilon},\rho_{\epsilon}) \]
Now we want to check the results
Model Comparison
Now we want to use the WAIC, DIC and MLIK to compare the models
Model check and comparison
Model Checking for Linear Models
In this exercise we will:
- Learn about some model assessments techniques available in INLA
- Conduct posterior predictive model checking
Libraries to load:
library(dplyr)
library(tidyr)
library(INLA)
library(ggplot2)
library(patchwork)
library(inlabru) Recall a simple linear regression model with Gaussian observations
\[ y_i\sim\mathcal{N}(\mu_i, \sigma^2), \qquad i = 1,\dots,N \]
where \(\sigma^2\) is the observation error, and the mean parameter \(\mu_i\) is linked to the linear predictor through an identity function:
\[ \eta_i = \mu_i = \beta_0 + \beta_1 x_i \] where \(x_i\) is a covariate and \(\beta_0, \beta_1\) are parameters to be estimated.
Simulate example data
We simulate data from a simple linear regression model
Code
beta = c(2,0.5)
sd_error = 0.1
n = 100
x = rnorm(n)
y = beta[1] + beta[2] * x + rnorm(n, sd = sd_error)
df = data.frame(y = y, x = x) Fitting the linear regression model with inlabru
Now we fit a simple linear regression model in inalbru by defining (1) the model components, (2) the linear predictor and (3) the likelihood.
# Model components
cmp = ~ -1 + beta_0(1) + beta_1(x, model = "linear")
# Linear predictor
formula = y ~ Intercept + beta_1
# Observational model likelihood
lik = bru_obs(formula = y ~.,
family = "gaussian",
data = df)
# Fit the Model
fit.lm = bru(cmp, lik)Residuals analysis
A common way for model diagnostics in regression analysis is by checking residual plots. In a Bayesian setting residuals can be defined in multiple ways depending on how you account for posterior uncertainty. Here, we will adopt a Bayesian approach by generating samples from the posterior distribution of the model parameters and then draw samples from the residuals defined as:
\[ r_i = y_i - x_i^T\beta \]
We can use the predict function to achieve this:
res_samples <- predict(
fit.lm, # the fitted model
df, # the original data set
~ data.frame(
res = y-(beta_0 + beta_1) # compute the residuals
),
n.samples = 1000 # draw 1000 samples
)The resulting data frame contains the posterior draw of the residuals mean for which we can produce some diagnostics plots , e.g.
Residuals checks for Linear Model
ggplot(res_samples,aes(y=mean,x=1:100))+geom_point() +
ggplot(res_samples,aes(y=mean,x=x))+geom_point()We can also compare these against the theoretical quantiles of the Normal distribution as follows:
QQPlot for Linear Model
arrange(res_samples, mean) %>%
mutate(theortical_quantiles = qnorm(1:100 / (1+100))) %>%
ggplot(aes(x=theortical_quantiles,y= mean)) +
geom_ribbon(aes(ymin = q0.025, ymax = q0.975), fill = "grey70")+
geom_abline(intercept = mean(res_samples$mean),
slope = sd(res_samples$mean)) +
geom_point() +
labs(x = "Theoretical Quantiles (Normal)",
y= "Sample Quantiles (Residuals)") Posterior Predictive Checks
Now, instead of generating samples from the mean, we will account for the observational process uncertainty by:
- Sampling \(y^{1k}_i\sim\pi(y_i|\mathbf{y})\) \(k = 1,\dots,M;~i = 1,\ldots,100\) using
generate()(here we will draw \(M=500\) samples)
samples = generate(fit.lm, df,
formula = ~ {
mu <- (beta_0 + beta_1)
sd <- sqrt(1 / Precision_for_the_Gaussian_observations)
rnorm(100, mean = mu, sd = sd)
},
n.samples = 500
) - Comparing some summaries of the simulated data with the one of the observed one
Here we compare (i) the estimated posterior densities \(\hat{\pi}^k(y|\mathbf{y})\) with the estimated data density and (ii) the samples means and 95% credible intervals against the observations.
# Tidy format for plotting
samples_long = data.frame(samples) %>%
mutate(id = 1:100) %>% # i-th observation
pivot_longer(-id)
# compute the mean and quantiles for the samples
draws_summaries = data.frame(mean_samples = apply(samples,1,mean),
q25 = apply(samples,1,function(x)quantile(x,0.025)),
q975 = apply(samples,1,function(x)quantile(x,0.975)),
observations = df$y)
p1 = ggplot() + geom_density(data = samples_long,
aes(value, group = name), color = "#E69F00") +
geom_density(data = df, aes(y)) +
xlab("") + ylab("")
p2 = ggplot(draws_summaries,aes(y=mean_samples,x=observations))+
geom_errorbar(aes(ymin = q25,
ymax = q975),
alpha = 0.5, color = "grey50")+
geom_point()+geom_abline(slope = 1,intercept = 0,lty=2)+labs()
p1 +p2GLM model checking
In this exercise we will:
- Learn about some model assessments techniques available in INLA
- Conduct posterior predictive model checking using CPO and PIT
Libraries to load:
library(dplyr)
library(INLA)
library(ggplot2)
library(patchwork)
library(inlabru) In this exercise, we will use data on horseshoe crabs (Limulus polyphemus) where the number of satellites males surrounding a breeding female are counted along with the female’s color and carapace width.
A possible model to study the factors that affect the number of satellites for female crabs is
\[ \begin{aligned} y_i&\sim\mathrm{Poisson}(\mu_i), \qquad i = 1,\dots,N \\ \eta_i &= \mu_i = \beta_0 + \beta_1 x_i + \ldots \end{aligned} \]
We can explore the conditional means and variances given the female’s color:
crabs <- read.csv("datasets/crabs.csv")
# conditional means and variances
crabs %>%
summarise( Mean = mean(satell ),
Variance = var(satell),
.by = color) color Mean Variance
1 medium 3.294737 10.273908
2 dark 2.227273 6.737844
3 light 4.083333 9.719697
4 darker 2.045455 13.093074
The mean of the number of satellites vary by color which gives a good indication that color might be useful for predicting satellites numbers. However, notice that the mean is lower than its variance suggesting that overdispersion might be present and that a negative binomial model would be more appropriate for the data (we will cover this later).
Fitting the model
First, lets begin fitting the Poisson model above using the carapace’s color and width as predictors. Since, color is a categorical variable in our model we need to create a dummy variable for it. We can use the model.matrix function to help us constructing the design matrix and then append this to our data:
crabs_df = model.matrix( ~ color , crabs) %>%
as.data.frame() %>%
select(-1) %>% # drop intercept
bind_cols(crabs) %>% # append to original data
select(-color) # remove original color categorical variableThe new data set crabs_df contains a dummy variable for the different color categories (dark being the reference category). Then we can fit the model in inlabru as follows:
cmp = ~ -1 + beta0(1) + colordarker +
colorlight + colormedium +
w(weight, model = "linear")
lik = bru_obs(formula = satell ~.,
family = "poisson",
data = crabs_df)
fit_pois = bru(cmp, lik)
summary(fit_pois)inlabru version: 2.13.0.9011
INLA version: 25.09.19
Components:
Latent components:
beta0: main = linear(1)
colordarker: main = linear(colordarker)
colorlight: main = linear(colorlight)
colormedium: main = linear(colormedium)
w: main = linear(weight)
Observation models:
Family: 'poisson'
Tag: <No tag>
Data class: 'data.frame'
Response class: 'integer'
Predictor: satell ~ .
Additive/Linear: TRUE/TRUE
Used components: effects[beta0, colordarker, colorlight, colormedium, w], latent[]
Time used:
Pre = 0.848, Running = 0.195, Post = 0.00852, Total = 1.05
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
beta0 -0.501 0.196 -0.885 -0.501 -0.117 -0.501 0
colordarker -0.008 0.180 -0.362 -0.008 0.345 -0.008 0
colorlight 0.445 0.176 0.101 0.445 0.790 0.445 0
colormedium 0.248 0.118 0.017 0.248 0.479 0.248 0
w 0.001 0.000 0.000 0.001 0.001 0.001 0
Deviance Information Criterion (DIC) ...............: 917.12
Deviance Information Criterion (DIC, saturated) ....: 561.74
Effective number of parameters .....................: 5.01
Watanabe-Akaike information criterion (WAIC) ...: 929.70
Effective number of parameters .................: 16.51
Marginal log-Likelihood: -489.43
is computed
Posterior summaries for the linear predictor and the fitted values are computed
(Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')
Model assessment and model choice
Now that we have fitted the model we would like to carry some model assessments. In a Bayesian setting, this is often based on posterior predictive checks. To do so, we will use the CPO and PIT - two commonly used Bayesian model assessment criteria based on the posterior predictive distribution.
The posterior predictive distribution for a predicted value \(\hat{y}\) is
\[ \pi(\hat{y}|\mathbf{y}) = \int_\theta \pi(\hat{y}|\theta)\pi(\theta|\mathbf{y})d\theta. \]
The probability integral transform (PIT) introduced by Dawid (1984) is defined for each observation as:
\[ \mathrm{PIT}_i = \pi(\hat{y}_i \leq y_i |\mathbf{y}{-i}) \]
The PIT evaluates how well a model’s predicted values match the observed data distribution. It is computed as the cumulative distribution function (CDF) of the observed data evaluated at each predicted value. If the model is well-calibrated, the PIT values should be approximately uniformly distributed. Deviations from this uniform distribution may indicate issues with model calibration or overfitting.
Another metric we could used to asses the model fit is the conditional predictive ordinate (CPO) introduced by Pettit (1990), and defined as:
\[ \text{CPO}_i = \pi(y_i| \mathbf{y}{-i}) \]
The CPO measures the density of the observed value of \(y_i\) when model is fit using all data but \(y_i\). CPO provides a measure of how well the model predicts each individual observation while taking into account the rest of the data and the model. Large values indicate a better fit of the model to the data, while small values indicate a bad fitting of the model
To compute PIT and CPO we can either:
ask
inlabruto compute them by setoptions = list(control.compute = list(cpo = TRUE))in thebru()function arguments.set this as default in
inlabruglobal option using thebru_options_setfunction.
Here we will do the later and re-run the model
bru_options_set(control.compute = list(cpo = TRUE))
fit_pois = bru(cmp, lik)Now we can produce histograms and QQ plots to assess for uniformity in the PIT values which can be accessed through inlabru_model$cpo$pit :
fit_pois$cpo$pit %>%
hist(main = "Histogram of PIT values")
qqplot(qunif(ppoints(length(fit_pois$cpo$pit))),
fit_pois$cpo$pit,
main = "Q-Q plot for Unif(0,1)",
xlab = "Theoretical Quantiles",
ylab = "Sample Quantiles")
qqline(fit_pois$cpo$pit,
distribution = function(p) qunif(p),
prob = c(0.1, 0.9))Both Q-Q plots and histogram of the PIT values suggest a not so great model fit. For the CPO values, usually the following summary of the CPO is often used:
\[ -\sum_{i=1}^n \log (\text{CPO}\_i) \]
This quantities is useful when comparing different models - a smaller values indicate a better model fit. CPO values can be accessed by typing inlabru_model$cpo$cpo.